Large induced forest in a planar graph.

Conjecture Every planar graph on verices has an induced forest with at least vertices.

This conjecture is best possible. (See [AW]). It follows from Borodin's theorem stating that every planar graph has an acyclic -colouring that every planar graph on verices has an induced forest with at least vertices. The conjecture holds for planar graph with girth at least , because they can be partitionned into a stable set and a forest [BG] (see also [KT]).